A Theoretical Study of Ultrafast Phenomena in Complex Atoms
A Doctoral Thesis by
Carl Leon Mikael Petersson
Supervised by
Fernando Mart´ın
Departamento de Qu´ ımica
Acknowledgements
My sincerest regards go out to all those who have supported me in the creation of this work.
Foremost, I would like to thank my supervisor, Fernando Mart´ın, for giving me the opportunity to write this work. It has been a great blessing to be able to work under such a prolific scientist, and in a group as prominent as the UAM XCHEM one. I have greatly enjoyed working at UAM and thank all my friends there for their support.
Luca Argenti, who acted as my secondary supervisor during the first few years, also has my deepest appreciation. Without his great support and patience, this work would not have been possible.
Both Carlos Marante and Markus Klinker, whom I have worked closely with, and who have been of great help during my studies, also have my deepest gratitude.
Furthermore, my gratitude goes out to Kilian Arteaga, Pedro Fern´andez, and especially Vicent Borr`as, for assisting me in writing the Spanish parts of my thesis.
I am also deeply grateful to the experimental groups we have cooperated with; at Lund, ETH, Universit´e Paris-Saclay, and The Ohio State University; who have offered great input, and helped ground our theoretical musings in reality.
Finally, I am eternally grateful to my family. Their help with proofreading
Abstract
The ultrafast movement of electrons is a driving force of chemical reactions, making it a highly desirable avenue for study. This thesis studies such move- ments, making use of pump-probe methods such as attosecond transient ab- sorption spectroscopy (ATAS) and reconstruction of attosecond beatings by interference of two-photon transitions (RABITT), in complex atomic systems.
The main approach used to solve the time-dependent Schr¨odinger equation (TDSE) was exact, attosecond, full-electron, ab-initio calculations.
Firstly, helium was probed above the second ionisation threshold, where several ionisation channels are open, using accurate ab-initio calculations. Here, the ATAS method was employed to predict beatings between the autoionising 3snp1Po resonances and nearby1Seand1De states. More surprisingly, two- photon beatings between the doubly-excited 3s3p state and the1Pocontinuum were also observed, demonstrating control of the correlated, two-electron, multi- channel wave packet.
Secondly, two studies of neon were carried out below the second ionisation threshold. The first makes use of ATAS calculations to probe beatings between the autoionising neon states. Using a two-colour, mixed extreme-ultraviolet (XUV) near-infrared (NIR) pump, one-photon beatings between the 2s−13p1Po and the nearby 2s−13s1Seand 2s−13d1Deresonances are observed. Further, one- and two-photon beatings between the autoionising 2s−13`, ` ∈ {0,1} and the1Po
vicinity of multiple resonances. This is far from trivial, and interferometric methods have until now been restricted to simpler energy-regions, due to the difficulty of accurately describing the electron correlation associated with the more complex case, making accurate ab-initio calculations needed to guide experiments unavailable. Despite the complex energy-dependence of the phase when several resonances are present, presented results from experiment and ab- initio theory are in excellent agreement. Further, using a simple extension of the Fano model for resonant continua, the contributions of the different involved resonances are disentangled. Such simple models are highly desirable in more advanced systems, where accurate ab-initio calculations are inaccessible.
The ab-initio results of both neon studies were carried out using the newly developed XCHEM methodology, which is thus further validated by the excel- lent agreement with presented experiments and previous studies.
Finally, a RABITT study of argon in the vicinity of the 3s−1n` resonances was performed. Angularly resolved, experimental results are presented, showing the anisotropy of the atomic phase in smooth continua as well as the vicinity of resonances. Due to the complexity of the system, no ab-initio results are present.
Instead, simpler interferometric models are used to successfully explain the anisotropic behaviour of the phase.
Resumen
El movimiento ultrarr´apido de electrones es la fuerza motriz de las reacciones qu´ımicas, por lo cual su estudio resulta muy atractivo. Esta tesis se dedica al estudio de ese tipo de movimientos, utilizando m´etodos de bombeo y sonda, como espectroscop´ıa de absorci´on transitoria de attosegundos (ATAS) y recon- strucci´on de ”beatings” de attosegundo por interferencia de transiciones de dos fotones (RABITT), en ´atomos complejos. El m´etodo principal utilizado para resolver de la ecuaci´on de Schr¨odinger dependiente del tiempo fue la propagaci´on exacta (ab-initio) considerando todos los electrones.
En primer lugar, se investig´o el ´atomo de helio por encima del segundo um- bral de ionizaci´on, donde existen varios canales de ionizaci´on. Aqu´ı, el m´etodo de ATAS se emple´o para predecir beatings entre las resonancias 3snp1Poy esta- dos1Sey1Decercanos. Sorprendentemente, tambi´en se observaron beatings de dos fotones, lo cual muestra control del paquete de ondas correlacionado multicanal de dos electrones.
En segundo lugar, dos estudios por debajo del segundo umbral de ionizaci´on del ne´on se llevaron a cabo. El primero utiliza c´alculos de ATAS para investigar los beatings entre estados autoionizantes de ne´on. Utilizando un bombeo de dos colores, radiaci´on ultravioleta extrema (XUV) mezclada con radiaci´on infrarrojo cercano (NIR), es posible observar beatings entre la resonancia del 2s−13p1Poy las 2s−13s1Sey 2s−13d1De. Adem´as, se predicen beatings de uno y dos fotones
El segundo usa el m´etodo de RABITT para estudiar la fase at´omica en las cer- can´ıas de las resonancias m ´ultiples. Hasta ahora, los m´etodos interferom´etricos han estado restringidos a regiones de energ´ıa de hasta una resonancia, a causa de las dificultades en llevar a cabo propagaciones exactas (ab-initio), las cuales dependen de la correlaci´on electr´onica para describir bien los experimentos.
A pesar de la complejidad de la dependencia de la energ´ıa con la fase, debido a la presencia de varias resonancias, los resultados te´oricos obtenidos comparan muy bien con los resultados experimentales presentados. Adem´as, usando una extensi´on del modelo de Fano para continuos resonantes, las contribuciones de las distintas resonancias se han podido resolver. Modelos m´as simples son necesarios en sistemas m´as avanzados, donde c´alculos ab-initio son inaccesibles.
Los resultados ab-initio presentados en ambos estudios se realizaron con el m´etodo XCHEM recientemente propuesto, dando as´ı validez al m´etodo.
Finalmente, se realiz´o un estudio RABITT cerca de las resonancias 3s−1n` del arg´on. Se presentan experimentos mostrando la dependencia angular de la fase at´omica, tanto en continuos suaves como en las cercan´ıas de resonancias. Debido a la complejidad del sistema, no se presentan resultados ab-initio. En cambio, mediante modelos interferom´etricos se ha podido explicar el comportamiento anis´otropo de la fase.
Abbreviations
APT Attosecondpulsetrain
AT Autler-Townes
ATAS Attosecondtransientabsorptionspectroscopy CAP Complexabsorptionpotential
CAS Completeactivespace DES Doublyexcitedstates FWHM Fullwidth athalfmaximum GABS CombinedGaussian–B-spline
HF Hartree-Fock
HH Highharmonic
HHG Highharmonicgeneration
IR Infrared
NIR NearInfrared
PAD Photoelectronangulardistribution PES Photoelectronspectrum
PI Parention
QCP QuantumChemistryPackage
RABITT Reconstruction ofatomicbeatings byinterference oftwo- photontransitions
SAE SingleActiveElectron
SB Sideband
SCF SelfConsistentField
TDPT Time-dependentperturbationtheory TDSE Time-dependentSchr¨odingerequation TISE Time-independentSchr¨odingerequation
VIS Visible
XUV Extremeultraviolet
List of Publications
Barreau, L.,Petersson C. L. M., M. Klinker, A. Camper, C. Marante, T. Gor- man, D. Kiesewetter, L. Argenti, P. Agostini, J. Gonz´alez-V´azquez, P.
Sali`eres, L. F. DiMauro, and F. Mart´ın (June 2019). “Disentangling Spectral Phases of Interfering Autoionizing States from Attosecond Interferometric Measurements”. In: Phys. Rev. Lett. 122 (25), p. 253203. doi: 10 . 1103 /PhysRevLett.122.253203. url:https://link.aps.org/doi/10.1103 /PhysRevLett.122.253203.
Cirelli, C., C. Marante, S. Heuser,C. L. M. Petersson, ´A. J. Gal´an, L. Argenti, S.
Zhong, D. Busto, M. Isinger, S. Nandi, et al. (2018). “Anisotropic photoe- mission time delays close to a Fano resonance”. In: Nature Communications 9.1, p. 955. issn: 2041-1723. doi:10.1038/s41467- 018- 03009- 1. url:
https://doi.org/10.1038/s41467-018-03009-1.
Petersson,C. L. M., L. Argenti, and F. Mart´ın (2017). “Attosecond transient absorption spectroscopy of helium above the N= 2 ionization threshold”.
In: Physical Review A 96.1, p. 013403. doi:10.1103/PhysRevA.96.013403.
url:https://journals.aps.org/pra/abstract/10.1103/PhysRevA.9 6.013403.
Contents
I Background 1
1 Introduction 3
2 Light-Matter Interaction 7
II Theory 13
3 Ab-Initio Theory 15
3.1 Attosecond Transient Absorption Spectroscopy . . . .16
3.1.1 The Transient Absorption Spectrum . . . 18
3.1.2 The Bidimensional Spectrum . . . 24
3.1.3 Features of the ATAS Spectrum . . . 26
3.2 Reconstruction of Attosecond Beatings by Interference of Two-Photon Transitions . . . .33
3.2.1 The Sideband Phase . . . 40
3.3 Ab-Initio Propagations . . . .47
3.3.1 Field-Free Propagation . . . 51
3.3.2 Field-Driven Propagation . . . 52
CONTENTS
3.3.3 Complex Absorption Potential . . . 58
3.4 Extracting Observables . . . .63
3.4.1 The Transient Absorption Spectrum . . . 64
3.4.2 The Photoelectron Spectrum . . . 70
4 Ab-Initio Bases 73 4.1 The B-Spline Helium Basis . . . .74
4.1.1 The Helium Eigenfunctions . . . 75
4.1.2 The B-Spline Basis . . . 77
4.1.3 Diagonalising the Hamiltonian . . . 80
4.2 The XCHEM Basis . . . .82
4.2.1 The Atomic Basis . . . 83
4.2.2 The GABS Hybrid Basis . . . 85
4.2.3 Diagonalising the Hamiltonian . . . 87
5 Simple Interferometric Models 89 5.1 Calculation of the Matrix Elements . . . .90
5.1.1 Transition Amplitudes Between Featureless Continua . . . 92
5.1.2 Fano Formalism and Resonant Transition Amplitudes . . . 93
5.2 Angularly Resolved Atomic Phases . . . .99
III Results 103
6 Helium 105 6.1 ATAS Above the N = 2 Threshold . . . .1056.1.1 The Cross Section . . . 108
6.1.2 The Attosecond Transient Absorption Spectrum . . . 109
CONTENTS
7 Neon 115
7.1 ATAS Below the Second Threshold . . . .116
7.1.1 The Cross Section . . . 118
7.1.2 The Attosecond Transient Absorption Spectrum . . . 119
7.1.3 The Bidimensional Spectrum . . . 121
7.1.4 Conclusions . . . 123
7.2 RABITT in the Vicinity of Multiple Resonances . . . .123
7.2.1 The Photoelectron Spectrum . . . 129
7.2.2 Extracted Phases . . . 131
7.2.3 Conclusions . . . 134
8 Argon 137 8.1 Anisotropic Time Delays Near a Resonance . . . .137
8.1.1 Delay-Integrated Asymmetry Parameters . . . 142
8.1.2 Delay-Resolved Asymmetry Parameters . . . 144
8.1.3 Angularly Resolved Atomic Delays . . . 145
8.1.4 Angularly and Spectrally Resolved Atomic Delays . . . 147
8.1.5 Ionisation Paths and Angle-Dependent Atomic Delays . . . 149
8.1.6 Conclusions . . . 153
IV Conclusions 155
9 Conclusions 157
10Conclusiones 161
V Appendices 165
CONTENTS
A Atomic Units 167
B Time-Dependent Perturbation Theory 169
C Numerical Accuracy and Efficiency 173
C.1 Accuracy of the Operator Split . . . .174
C.1.1 The Strang Splitting . . . 175
C.1.2 Addition of the CAP . . . 176
C.2 Time Complexity . . . .176
C.2.1 Field-Free Propagation . . . 177
C.2.2 Field-Driven Propagation . . . 178
C.2.3 Complex Absorption Potential . . . 180
Bibliography 180
List of Figures
3.1 A schematic view of the ATAS field . . . 17
3.2 An example of the ATAS spectra . . . 25
3.3 A schematic view of ATAS beatings . . . 29
3.4 A schematic view of slow-varying ATAS fringes . . . 31
3.5 The three-step model of HHG . . . 36
3.6 The composition of the RABITT Field . . . 36
3.7 The RABITT spectrum . . . 39
3.8 The scattering phase and Wigner time delay . . . 46
3.9 The eigenstates of the quenched Hamiltonian . . . 61
3.10 The numerical dipole evolution with and without filters . . . 62
3.11 The short- and long-ranged TAS components . . . 66
3.12 The contributions of the different channels to the PES . . . 70
4.1 The difference between B-Splines of different order . . . 78
4.2 The XCHEM GABS basis . . . 86
6.1 A schematic view of the helium region studied using ATAS . . . 107
6.2 The helium cross-section above the N = 2 threshold . . . 108
6.3 The ATAS spectrum above the N = 2 threshold of helium . . . . 110
6.4 The ATAS spectrum above the N = 2 threshold of helium, with the background removed . . . 110
6.5 The bidimensional ATAS spectrum of helium . . . 111
LIST OF FIGURES
7.1 A schematic view of the neon region studied using ATAS. . . 118
7.2 The transient neon cross-section. . . 119
7.3 The ATAS spectrum of neon . . . 120
7.4 The bidimensional ATAS spectrum of neon . . . 122
7.5 A schematic view of the neon region studied using RABITT . . . 125
7.6 The neon RABITT spectrum . . . 128
7.7 The one-photon photoelectron yield of HH63 . . . 130
7.8 The RABITT atomic phase difference of SB62to SB64 . . . 131
7.9 The resonant contributions to the atomic phase difference of SB64135 8.1 The spectral distributions of the experimental RABITT pumps . 140 8.2 The argon RABITT spectra . . . 141
8.3 Angular photoelectron distributions and βiparameters . . . 142
8.4 The time-dependence of the β2parameters. . . 144
8.5 The angularly resolved emission delays. . . 146
8.6 Emission delays resolved by energy and angle . . . 148
8.7 Angular dependence of matrix element amplitudes and phases . 150 8.8 Angular dependence of argon continuum-continuum phases . . 152
I
Background
Background
Chapter 1
Introduction
The nature of chemical reactions is largely governed by the motions of bound electrons, making an increased understanding of electron movement in the proper time-scale a highly desirable objective. The movement – referred to as ultrafast – of atomic valence electrons tends to occur in the order of magnitude of attoseconds or a few femtoseconds. Such phenomena can be studied by employing what is known as the pump-probe approach: This approach makes use of two separate laser pulses to interact with the studied system: A high- energy, often extreme ultraviolet (XUV), pulse, referred to as a pump pulse, is used to ionise an atomic system. A second pulse, known as a probe or control pulse, can subsequently be used to extract information about the state of the excited atom. By varying the delay between the two pulses, a time-dependent view of the atom emerges.
These methods have only recently been extended to the domain of attosec- ond physics, creating a great demand for theory. As it is not yet possible to experimentally extract relevant information without the proper guidance of theory, theory and experiment have to be developed in parallel.
CHAPTER 1. INTRODUCTION
(RABITT). Both proven themselves capable tools for obtaining information not available outside the field of attosecond science. Furthermore, compared to the third, streaking, both ATAS and RABITT are more wildly used and are associated with a larger amount of available experimental literature data.
ATAS is an extension of transient absorption spectroscopy (TAS), where the probe pulse is introduced in order to modify the absorption spectrum. Although its femtosecond analogy has been in use for over 50 years, it is only recently (Goulielmakis et al.2010) pump-probe TAS has been extended to the attosecond domain. ATAS is able to track time-resolve electron dynamics and reconstruct electron wave packets, and RABITT enabling measurements of photoemission delays in the order of hundreds of attoseconds.
RABITT was originally (Agostini et al.2004; Paul et al.2001) proposed to study the pulses generated by high-harmonic generation (HHG) (Ferray et al.
1988; McPherson et al.1987), by focusing on smooth, featureless continua.
Once the characteristics of one such pulse is known, however it can instead be used to study more complex atomic and molecular regions. With that, RABITT has enabled measurements of photoemission delays in the order of hundreds of attoseconds.
For atoms more complex than hydrogen, the electron movement is a many- body problem, and theoretically impossible to describe exactly. Instead approx- imative methods need to be employed. Unfortunately, simpler methods, such as the Hartree (1935)-Fock (1930) (HF) self-consistent field (SCF) approach or the single-active electron (SAE) approximation, which reduce the problem by only considering a single electron at a time, fail to accurately account for the much more intertwined electron-electron interaction – what is known as electron- correlation (H¨attig et al.2011). For this reason, more elaborate descriptions are necessary in order to accurately model these systems.
This thesis considers the theory behind the ultrafast movement of elec- trons in three complex atoms: helium, neon, and argon. These systems are modeled using accurate, full-electron, ab-initio pump-probe calculations; their description enabled through approaches such as a K-matrix (Argenti et al.2006;
Lindroth et al.2012) B-spline (Argenti et al.Unpublished) method and the XCHEM (Marante et al.2014,2017a,b) method. Although the systems consid- ered here are all noble gas atoms, and can thus hardly be considered reactive, the underlying theory may well be extended to more reactive atomic and molecular systems.
The underlying document is structured as follows: PartI, to which the present chapter pertains, make up the introduction. It also contains chapter2, in which a basic description of quantum physics, and the interaction between light and atoms, is given. In partII, the main bulk of the theory is described.
It consists of three chapters. Firstly, chapter3 contains information on the specific experimental, attosecond pump-probe methods employed, on the nu- merical ab-initio propagation used to simulate them, and on how observables are extracted from the system. Secondly, chapter4contains information on the bases used during ab-initio propagations. Thirdly and lastly, chapter5 describes an extension of the Fano (1961) model to two-photon transitions, used to extract information about atomic systems without the need for expensive ab-initio calculations. In partIII, the results are given, divided into one chapter for each studied atomic system. Chapter6considers helium, chapter7neon, and chapter8argon. PartIVconsists of chapters9and10, which, in English and Spanish respectively, contain a summary of the conclusions of this work.
Finally, three appendices are included, located in partV. In appendixA, atomic units are discussed; in appendixB, time-dependent perturbation theory (TDPT) is described; and in appendixCthe split accuracy and time complexity of the ab-initio propagation operator is treated.
Throughout this thesis, Hartree (1928) atomic units (au) will be used unless otherwise stated. These are defined by setting the reduced Plank constant ~, the electron mass methe elemental charge e, and the Coulomb constant 4π10 (where
0is the vacuum permittivity) equal to one (~ = me= e = 4π0= 1). This unit system is convenient, as it greatly simplifies many relevant formulas. Several
Chapter 2
Light-Matter Interaction
One of the fundamental concepts that distinguishes quantum mechanics from its classical forerunner, is that it does away with the concept of determinism.
According to quantum mechanics, the outcome of an experiment can never be predicted with certainty, but only as a probability distribution. Instead of a particle having a fixed position, energy, and location, these quantities may be distributed over several values, their probability being described as a wave- function distribution. Each such quantity a which can be measured – each observable – is associated with a linear operator ˆA. This operator can be used to extract the expectation value of the quantity as
aexp = D ˆAE ≡ DΨ ˆA ΨE, (2.1) for a particle in a state ΨE
, which has been denoted using Dirac’s (1939) bra- ket notation: In this notation, a state may be described by a ket ΨE
, which is analogous to a column vector in linear algebra. Similarly, and analogously to a row-vector, the braD
Ψ is the Hermitian transpose of ΨE
. Bra-ket notation is
CHAPTER 2. LIGHT-MATTER INTERACTION
Unlike particles in Newtonian mechanics, the movement of quantum me- chanical waves are described by the time-dependent Schr¨odinger (1926) equa- tion (TDSE),
i∂
∂t ψ (t)E
= ˆH(r,t) ψ (t)E
, (2.2)
which considers the evolution in time t, of a state ψ (t)E
; where r is the position and the Hamiltonian operator ˆH is the sum
H = ˆˆ T + ˆV (2.3)
of the kinetic energy operator ˆT , and the potential energy operator ˆV. Equation 2.2can be solved by defining the time-evolution operator
U (r;tˆ 1, t0) = ˆTo
( exp
"
−i Z t1
t0
dt ˆH(r,t)
#)
(2.4)
between the times t0and t1, where the time-ordering operator ˆTo works on the expression within the brackets.
For a single particle, the kinetic energy operator ˆT can be written as T =ˆ 1
2ˆp2, (2.5)
where ˆp is the momentum operator. The operator ˆp may be written in terms of the gradient operator ˆ∇ =
∂x, ∂y, ∂z|
as ˆp = −i ˆ∇. The potential energy operator ˆV, on the other hand, can be described as a time and space-dependent scalar operator.
For a time-independent Hamiltonian ˆH, the Hamiltonian eigenfunctions ψ (r, t) can be written as standing waves
ψ (t)E
= ψE
exp(−iεt) (2.6)
of energy ε, by separating the time-dependent component from the functions
ψE
, which also are Hamiltonian eigenfunctions. This allows equation2.2to be simplified to the eigenvalue problem
ε ψ (t)E
= ˆH(r,t) ψ (t)E
, (2.7)
known as the time-independent Schr¨odinger equation (TISE). This mirrors the classical relation ε = εkin+ εpot, according to which the total energy can be seen as the sum of the kinetic (εkin) and the potential (εpot) energy.
In particular, this thesis consider atomic systems – systems consisting of a single nucleus surrounded by several electrons. The simplest atomic system, hydrogen, is described by the Hamiltonian
Hˆ0 = 1
2ˆp2 − 1
kˆrk (2.8)
in the inertial reference frame of the atomic core, positioned at the origin.
The second term – the potential energy – considers the Coulomb attraction between the core and the electron. This is a potential well, to which Hamiltonian eigenfunctions with negative energies can be considered bound.
The bound eigenfunctions can be divided, by the projectionD r ψn`mE
onto the position eigenstates rE
, as Dr ψn`mE
= Rn`(r)Y`m( ˆr), (2.9) into a radial component Rn`(r) (where r = krk) from the core, and an angular component Y`m( ˆr) (where ˆr = r/r), known as a spherical harmonic. They are defined by three integer quantum numbers: The principal quantum number n, the angular momentum `, and the magnetic quantum number m. These are restricted by n ≥ 1, 0 ≤ ` < n, and |m| ≤ `. The associated eigenenergies depend
as 1
CHAPTER 2. LIGHT-MATTER INTERACTION
Turning to larger atomic systems, the case is not as simple. For an atom with the atomic number Z, the Hamiltonian may be written as
Hˆ0 = − X
i
1
2ˆp2i − X
i
Z
jˆri + X
i, j>i
1
ˆri− ˆrj , (2.11) where ˆpi and ˆri are the ith electron momentum and position operator, and the third term has been introduced to account for electron-electron repulsion.
When more than one electron is present, the TISE has no exact solution, and a numerical approach must be employed.
A state ket Ψ (t)E
is expressible as a linear combination Ψ (t)E
= X
i
ci(t) ψiE
(2.12)
of Hamiltonian eigenstates. If the Hamiltonian is time-independent, the con- stants ci(t) vary as ci(t) = ci(0) exp(−iεit). When considering interaction be- tween an atom and an external, time-dependent light field; it is therefore convenient to describe the system using the eigenfunctions of the static, atomic Hamiltonian ˆH0, writing the total Hamiltonian as the sum
H(t) = ˆˆ H0 + ˆHI(t) (2.13)
of ˆH0and a time-dependent interaction component ˆHI(t). The interaction with the electric field can be modeled using the minimal-coupling Hamiltonian
H(t) =ˆ X
i
1
2 ˆpi+ αA(ˆri, t)2 − Φ (ˆri, t) + ˆVatom(ˆri), (2.14) where ˆVatom(ˆr) is the atomic potential, α is the fine-structure constant, and the field is defined by the vector potential A(r,t) and the scalar potential Φ (r,t).
The electric field can be written as
E (r, t) = − ∂A (r, t)
∂t − ∇Φ (r,t) (2.15)
in terms of A(r,t) and Φ (ˆr,t). As in the case for the global state phase, neither A (r, t) nor Φ (ˆr, t) are physical observables. Indeed, the Schr¨odinger equation is invariant under the gauge transformation
A (r, t) → A (r,t) + ∇χ (r,t) ∂
∂tχ Φ (r, t) → Φ (r,t) −1
c
∂
∂tχ (r, t) , Ψ (r, t) → Ψ (r,t) exp(−iχ (r,t)) ∂
∂tχ
(2.16a) (2.16b) (2.16c)
where the scalar function χ (r,t) has been introduced and c is the speed of light.
One useful gauge is the Coulomb gauge, defined by setting ∇ · A(r,t) = 0 and Φ (r, t) = 0. In this gauge, A (r, t) and ∇ commute – that is, the commutator h ˆA, ˆBi = ˆA ˆB − ˆB ˆA equals the null operator ([A(r,t),∇] = ˆ∅). For this reason, the expression
HˆI(t) = X
i
−iαA(ˆri, t)· ˆpi + 2α2
2 A2(ˆri, t) (2.17) is valid for the interaction Hamiltonian. In order to further simplify this expression, presently consider the external field. The total field can be written as a linear combination of linearly polarised, monochromatic fields. Each such monochromatic field of wavelength λ and frequency ω can be written as
A (r, t) = A0cos ˆn ·r
λ − ωt + φ
, (2.18)
where the vector A0is orthogonal to the direction ˆn of the polarization. When
CHAPTER 2. LIGHT-MATTER INTERACTION
For comparison, the shortest wavelengths considered here are in the order of magnitude of 300 au – several hundred times larger than the Bohr radius a0 (the average distance between the core and the electron in a hydrogen atom).
Two more gauges may now be considered. Firstly, the G¨oppert-Mayer trans- formation, defined by χ (t) = − ˆd · A(t), where ˆd =P
iˆri, allows for transforma- tion into the length gauge. In this gauge, the interaction Hamiltonian may be simplified to the expression
HˆLI = E (t) · ˆd, (2.19)
where ˆd is known as the length-gauge dipole operator. Throughout this thesis, however, the velocity gauge is used. It has the interaction Hamiltonian
HˆVI = αA(t) · ˆP , (2.20)
where ˆPis the velocity-gauge dipole operator, and can be arrived at through the gauge transformation χ (t) =N2eRt
−∞dt0A2(t0), where the sum over the electrons is carried out by multiplication of the total number Neof electrons.
This thesis only considers only linearly polarised light. Taking the ˆz axis to be the polarisation direction, equation2.20can be further simplified as
HˆVI = αA(t) ˆP , (2.21)
by defining ˆP = ˆP · ˆz and A(t) = A(t) · ˆz. The state may now, as suggested in equation2.12, be described as a linear combination
Ψ (t)E
= X
i
ci(t) ψiE
exp(−iεit) (2.22)
of Hamiltonian eigenstates ψiE
, with corresponding eigenenergies εi. The values ci(t) remain constant unless an external field is present, redistributing the state population. It is these basic process which are examined by the time- resolved studies discussed in this work.
II Theory Theory
Chapter 3
Ab-Initio Theory
The focus of this work lies on the study of time-dependent electron dynamics in atoms, induced by linearly polarised ultrashort light. Time-dependent electron dynamics, generally referred to as ultrafast phenomena, take place on time- scales on the order of several attoseconds or a few femtoseconds.
Resolving spectra on such time-scales is not without complications. One problem which arises stems from what is known as the Heisenberg uncertainty principle. Although also applicable to other observables, for time- and energy- operators it can be written as
σε· σt≥1
2 (3.1)
where σεis the uncertainty (here defined as the variance) in the energy determi- nation, and σtis that of the time before the system decays, at which point it can be measured (Svanberg1991).
This imposes strict limitations on the maximum possible resolutions of energy and time. As an illustrative example, demanding a resolution corre- sponding to an energy-uncertainty as low as σε= 7 · 10−3au – one tenth of the
CHAPTER 3. AB-INITIO THEORY
σt≈ 1730as, an uncertainty more than twenty times the time-resolution used in said study!
How, then, can such a precision be achieved? The answer to this question lies in the pump-probe nature of the methods employed in this work. Using two independent laser pulses, it is not so much the time, t, which is measured, but rather the time-delay, τ, between said pulses, which is varied to probe time- dependent dynamics, overriding the constraint imposed by the Heisenberg uncertainty principle. (Pollard et al.1992).
When considering, say, a two-photon absorption process (one photon associ- ated with each pulse), it is not the exact time between the two absorption events that is measured. The times when these events occur are not known – only their time probability distributions, determined by the respective temporal profiles of the pulses. It is thus the entire probability-distributions that are shifted with τ, which can be done with arbitrary precision.
This chapter is dedicated to the theory behind these methods, and the phe- nomena they probe. Two pump-probe methods are considered: The first, known as attosecond transient absorption spectroscopy (ATAS), is described in section 3.1, and the second, reconstruction of attosecond beatings by interference of two-photon transitions, in section3.2. These phenomena were theoretically modelled, using accurate, full-electron, numerical ab-initio propagations. The theory behind those propagations is discussed in section3.3(although the de- tails behind the basis employed during propagation are given in chapter4).
Finally, section3.4complements this discussion by describing how observables can be extracted from the ab-initio propagation.
3.1 Attosecond Transient Absorption Spectroscopy
The first probing technique utilised in this thesis is what is known as attosecond transient absorption spectroscopy (ATAS). ATAS was initially used by Gouliel- makis et al. (2010) to monitor the ultrafast dynamics involved in the coherent interference of valance-holes in krypton. It has since been used to monitor dynamics resulting from processes such as superpositions of singly excited states in helium (S. Chen et al.2012,2013a,b; Chini et al.2014; Herrmann et al.
2013), and in neon (Beck et al.2014; Ding et al.2016; Wang et al.2013), of doubly excited states in helium (Argenti et al.2015; Ott et al.2014), the buildup of Fano (1961) profiles in helium (Kaldun et al.2016), and hole alignment in neon (Heinrich-Josties et al.2014). Through this, it has established itself as a trusted tool for monitoring ultrafast atomic as well as molecular processes (Argenti et al.2015; Beck et al.2014; Goulielmakis et al.2010; Ott et al.2014).
The general concept of ATAS becomes evident when considering the pump- probe setup, illustrated in figure3.1. The system under observation is excited by what is typically a short, weak, extreme ultraviolet (XUV), pulse (shown in blue in figure3.1), intended to populate the studied energy-range of the spectrum. A second probe pulse is then used to probe the dynamics of the excited spectrum.
By varying the time-delay, τ, between the pulses, the time-evolution of the populated-spectrum, and the corresponding dynamics, can be studied.This
t E (t)
τ
CHAPTER 3. AB-INITIO THEORY
thesis considers the case of an infrared (IR) or visible (VIS) probe (shown in fully drawn red in figure3.1), stronger than the XUV, but too weak to further drain the ground state.
It should be noted that, using a weak XUV pump as described above, only states which are dipole-connected to the ground state are populated. Although states which are not dipole connected to the ground state may be involved in the dynamics probed by the probe pulse, the time-dependent behaviour of such states is not observed. If such processes are to be studied, however, a low-energy component (shown in dashed red in figure3.1) can be added to the pump (Ding et al.2016). This component redistributes the population over the excited states – including those not connected to the ground state via one-photon transitions.
In this manner, the field-free evolution of such states over the interval between the pump and the probe also becomes relevant. This is already a common practice in femtosecond pump-probe spectroscopy (Petersson et al.2017).
This work considers both the case where the pump does (see chapter7.1), and does not (see chapter6.1), include a low-frequency component.
3.1.1 The Transient Absorption Spectrum
In ATAS, the transient absorption spectrum (TAS) is observed. In order to understand the significance of the TAS, and how it can be extracted, consider a beam of light passing through a target medium, along the x−axis. Assuming that the target concentration is uniform, the proportion of light absorbed per unit of length will be constant. That is, when passing through a small interval
∂x, the relation
∂I (ω; x)
I (ω; x) = −c(ω)∂x, (3.2)
where I (ω;x) is the intensity of the field at frequency ω after having penetrated a distance of x into the medium, and c (ω) is the absorbance per length, holds.
This relation is known as the Beer-Lambert law (Svanberg1991). Dividing c (ω) with the number of atoms per unit of length, n, gives an effective atomic cross
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
section
σ (ω) =c (ω)
n , (3.3)
allowing equation3.2to be rewritten as
∂I (ω; x)
I (ω; x) = −σ (ω)n∂x. (3.4)
Solving equation3.4gives
I (ω; x) = I (ω; 0) exp [−σ (ω)nx], (3.5) or, equivalently,
σ (ω) =− 1 nxln
"
I (ω; x) I (ω; 0)
#
. (3.6)
Using tilde to denote the Fourier transform; the first order Taylor-expansion of equation3.6gives
σ (ω) = 1 n x
I (ω; 0)− I (ω;x) I (ω; 0) = 1
n x
˜E (ω; 0) 2− ˜E (ω; x) 2
˜E (ω; 0) 2 , (3.7)
where E (t;x) is the electric field, and the relation I (ω) = ˜E (ω) 2between the intensity and the norm of the electric field has been used. It is the cross-section σ (ω) which is taken as the TAS-spectrum.
CHAPTER 3. AB-INITIO THEORY
In order to derive an expression for σ (ω), consider Maxwell’s equations, written as
∇· D = 01
c (3.8a)
∇· B = 01
c (3.8b)
∇× E = −1 c
∂
∂tB (3.8c)
∇×H = 1 c
∂
∂tD (3.8d)
in vacuum (where the current-density is zero and no free charges are present), where ∇ is the gradient vector operator, D is the electric displacement, B is the magnetic field, and H is the magnetizing field. Assuming the magnetisation can be neglected, the relations
D = E + 4πP (3.9a)
H = B , (3.9b)
where P is the polarisation, follow directly from the definitions of D and H.
Equations3.8a,3.8c,3.8d,3.9a, and3.9b, along with the property
∇× (∇ × v) = ∇(∇ · v) − ∇2v (3.10) of the cross product, valid for any arbitrary vector v, allows the equation
∂2
∂t2D = 4πc2∇(∇ · P ) + c2∇2E (3.11) to be derived. Together with the second time derivative of equation3.9a, this gives the expression
∇2− 1 c2
∂2
∂t2
! E = 4π
c2
∂2
∂t2P − 4π∇(∇ · P ) (3.12)
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
for the wave-propagation.
Additional simplifications of this expression can be made by considering the limitations of the problem studied in this thesis. Firstly, only linearly polarised light – traveling in a predetermined direction here taken as convention to be the ˆx axis – is considered, allowing for the simplification
E (t; x) = E (t; x) ˆz (3.13)
to be made, splitting the electric field into the normalised z-axis vector ˆz and a scalar quantity E (t;x).
Further, the systems treated using this method are helium and neon – both noble gases having an isotropic ground-state. For this reason, the polarisation P (t) = 0 prior to the arrival of the field. Thus, the only polarisation present will be in the direction of the field, and writing an expression for P analogous to equation3.13it can be seen that
P =P (t; x) ˆz⇒ ∇ · P = 0. (3.14) With this in mind, equation3.12can be rewritten as
∂2
∂x2− 1 c2
∂2
∂t2
!
E (t; x) =4π c2
∂2
∂t2P (t; x) (3.15)
in the direction of the field.
Using the expressions
E (t; x) = 1
√2π Z∞
0 dω ˜E (ω; x) exp [iω (t− x/c)] + c.c. (3.16) and
P (t; x) = 1
√ Z∞
dω ˜P (ω; x) exp [iω (t− x/c)] + c.c. (3.17)
CHAPTER 3. AB-INITIO THEORY
of the Fourier-transform, where c.c. denotes the complex conjugate, the fre- quency distribution of equation3.15,
∂2
∂x2− 2iω c
∂
∂x
!
˜E (ω;x) = −4πω2
c2P (ω; x) ,˜ (3.18) can be derived (Santra et al.2011).
A third and final limitation must now be introduced. In this work, it is the atom itself, rather than a thick cluster or solid of atoms, that is studied.
Thus, the case of a thin, rather than a thick, medium is considered, and the field can be considered as varying slowly during one wavelength, allowing the second-order x-derivative in equation3.18to be neglected. This allows the further simplification
∂
∂x ˜E (ω;x) = −2πiω
cP (ω; x) ,˜ (3.19)
to be made. Due to the diluted nature of the observed medium, the polari- sation ˜P (ω; x)≈ ˜P (ω) can be approximated as constant, allowing the linear approximation
˜E (ω;x) = ˜E (ω;0) − 2πiω
cP (ω) x,˜ (3.20)
to be made. Inserting this in equation3.7gives
σ (ω) =−1 n
4πω
c Im( ˜P(ω)
˜E (ω) )
. (3.21)
Since the polarisation corresponds to the total dipole moment, ˜d (ω), of the n atoms considered, the further simplification
σ (ω) =−4πω
c Im( ˜d(ω)
˜E (ω) )
, (3.22)
can be made. This formula can be directly used to calculate the atomic spectrum (Argenti et al.2015).
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
It is possible, however, to further simplify this expression in the velocity gauge. In order to do so, consider the relation
∂
∂tD ˆAE = iDh ˆH(t), ˆAiE, (3.23) where ˆH(t) is the total Hamiltonian, is valid for any given time-independent operator ˆA (Ohl´en2005). In the velocity gauge, the Hamiltonian can be written as
H(t) = ˆˆ H0+ αA(t) · ˆP , (3.24) where ˆH0is the field-free atomic potential, α is the fine-structure constant, and the vectors A(t) and ˆP =PNe
i=1Pˆi contains in all directions, respectively, the field vector potential and the electron canonical moment operator for all Ne
electrons. The total dipole momentum operator vector, ˆd =PNe
i=1ˆri, contains the position vector operators ˆri for all electrons. Thus, in the velocity gauge, equation3.23gives
∂
∂tD ˆdE
= iD h ˆH0+ αA(t) · ˆP , ˆdi E
=D
− ˆPE +D
− αNeA (t)E
. (3.25) Hence, it can be seen that the dipole moment, written as
∂
∂td (t) =−p (t) − αNeA (t) (3.26) in the direction of the field, can be divided into two components – one corre- sponding to the electron canonical momentum and the other to the external field. The corresponding frequency-domain expression is
d (ω) =˜ i
ω˜p (ω) +iαNe
ω A (ω) .˜ (3.27)
CHAPTER 3. AB-INITIO THEORY
be inserted into equation3.22. As the fraction corresponding to the external- field component to the dipole is real, the expression
σ (ω) =4π ωIm
(˜p (ω) A (ω)˜ )
(3.29)
is arrived at. This is the expression which has been used to extract the TAS in this work.
A numerical treatment of the TAS can be found in section3.4.1. One point from that section which bears repeating here, is that only the dipole response with the ground state is significant enough to be visible in the TAS. For this reason, only states which are dipole-coupled with the ground state are directly observed1.
By varying the delay τ between the pulses, a time-resolved TAS σ (ωr;τ) can be defined. To provide an illustrative example, parts of such a spectrum, for the case described in chapter7.1, is shown in figures3.2aand3.2c. The system considered is neon. For the energy ranges discussed herein, a 2p valence electron can either be excited to ionise said atom, or either one 2s or two 2p electrons can be excited to give rise to an autoionising resonance (Barreau et al.
2019). This gives rise to a continuum containing several features.
In this case, neon is studied with a mixed XUV-NIR (near infrared) pump and an NIR probe of frequency ωNIR= 0.05879 au. The corresponding spectrum when no probe pulse is used is denoted as σ0(ωr) below.
3.1.2 The Bidimensional Spectrum
In order to further study the ATAS spectrum, it is also useful to consider the changes to σ (ωr, τ) with τ in the frequency-, rather than the time-, domain.
1In all communications making up part of this thesis which makes use of ATAS, the ground state is of symmetry1Se. In such cases, only1Postates are directly visible.
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
1.54 (a) ε3s− ωNIR
ε3p− 2ωNIR
1.56
τα τβ
(c)
-750 0 750 1500 2250
1.66
ε3p
1.68
τ [au]
−1.5 arb. u. −1.0 arb. u.
(b)
(d)
0 2ωNIR 2ωNIR
ωτ[au]
−1.0 arb. u. 5.0 arb. u.
∼ ε3s ωr[au]
Figure 3.2: The natural logarithm of part of the TAS ln{σ (ωr, τ)} ((a)and(c)) and the bidimensional spectrum ln{ ˜σ (ωr, ωτ)} ((b)and(d)) discussed in chapter7.1Two energy ranges are displayed, one near the 2s−13p (labeled 3p) resonance ((c)and(d)) and one at lower energies where one- and two-photon beatings (illustrated to the right with arrows) with the 2s−13s (labeled 3s) and 2s−13p resonances respectively can be found ((a)and(b)). The time-delays are divided into three time-delay intervals using the markers ταand τβThe nearby 2p−23s3p resonance was excluded from the calculations, providing a clear view of the 2s−13p resonance.
This is done via the Fourier transform
˜
σ (ωr, ωτ) = Z
dτh
σ (ωr, τ)− σ0(ωr)i e−iωττ
√2π (3.30)
of the TAS. Here the background component σ0(ωr) has been removed. To un- derstand why, consider that when integrating over τ ∈ (−∞,∞), this component corresponds to a Dirac delta function σ0(ωr)δ0(ωτ); and that when for numeri- cal reasons integrating over a finite interval τ ∈ (−τmin, τmax), it corresponds to the Fourier transform σ0(ωr)Fn
θ (τ− τmin) − θ (τ + τmax)o
of a plateau, where θ (x) is the Heaviside step function. In the former case, the contribution of (ω ) is trivial; in the latter, it is unphysical. The Fourier transform in equa-
CHAPTER 3. AB-INITIO THEORY
3.2d.
As the XUV pump is weak, the frequency-distribution of the dipole response can be written as
d (ω˜ r) = Z
dωeχ (ωr, ωe) ˜EXUV(ωe), (3.31) where ˜EXUVis the XUV-component of the field, χ (ωr, ωe) is the electric suscep- tibility of the dressed atom, and ωe is used to denote the excitation energy (Argenti et al.2015). From this, it can be seen that the non-diagonal compo- nent χnd(ωr, ωe) of χ (ωr, ωe) relates (Argenti et al.2015) to the bidimensional spectrum as
˜
σ (ωr, ωτ) =(2π)3/2ωr ic
χnd(ωr, ωr− ωτ) − χnd(ωr, ωr+ ωτ)
, (3.32) where c is the speed of light, the diagonal component having already been removed with the removal of σ0(ωr) in equation3.30.
3.1.3 Features of the ATAS Spectrum
The time-delay domain of the TAS can be divided into three different intervals:
Using the notation in figure 3.2, these are τI− = (−∞,τα], τI0 = τα, τβ
, and τI+=h
τβ,∞
. The regions are selected so that the probe and the XUV component of the pump significantly overlap for τ ∈ τI0.
Large, Negative Time-Delays
Consider first the case of τ ∈ τI−. This region is characterised by the probe arriving before the pump without the two overlapping. Assuming the probe is too weak to not significantly drain the ground state, it will not have any significant effect. Because of this, the relation
τ∈ τI−⇒ σ (ω;τ) = σ0(ω) (3.33) is valid in said region.
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
Near-Zero Time-Delays
Next, there is the time interval above dubbed τI0. In this region, the atomic system is dressed during excitation.
One phenomenon in particular which was observed as part of this thesis (in Petersson et al. [2017], discussed in chatper6.1) in this region is the the special case of the Stark effect known as Autler-Townes (1955) (AT) splittings.
Although the phenomenon is not clearly visible in figure3.2, it has previously been observed during ATAS (Argenti et al.2017; Wu et al.2016).
AT splittings corresponds to a splitting of the resonance response energy, resulting from the transition between two resonances by a strong (IR, in the case of ATAS) field. This can be illustrated with a very simple model, by considering a two-level system, with two eigenstates, ψαE
and ψβE
, with complex populations cα and cβ, their energies ωα and ωβ being separated by an energy ωαβ (Wu et al.2016): Ignoring sub-cycle interactions (what is known as the rotating wave approximation), the time-dependent Schr¨odinger equation (TDSE) can be written as
∂
∂t
cα cβ
=
0 Ω(t;τ)2 exp(i∆t)
Ω(t;τ)
2 exp(−i∆t) 0
cα cβ
, (3.34)
where ∆ = ωIR− ωαβis the detuning and, denoting the total dipole operator ˆd and the dressing IR field EIR(t;τ),
Ω (t; τ) = EIR(t;τ)D
ψα ˆd ψβE
(3.35) is known as the Rabi-frequency. Approximating the field as constant (with a constant Rabi-frequency Ω0= Ω(t;τ)), this can be solved for cα, giving the result
cα(t) ∝Ωe− ∆ 2eΩ exp
−iΩ+∆e2 t
e , (3.36)
CHAPTER 3. AB-INITIO THEORY
where eΩ = q
Ω20+ ∆2.
Thus, the population of ψαE
is modified by the IR as
cαexp(−iωαt)→Ωe− ∆ 2eΩ cαexp
−i
ωα−Ω+∆e2 t
+Ω + ∆e 2eΩ cαexp
−i
ωα+Ω−∆e2 t
, (3.37)
splitting the undressed energy eigenvalue into two distinct dressed energy components.
Large, Positive Time-Delays
Finally, consider the interval of τ ∈ τI+, where the probe arrives after the pump without overlapping. The structure of this region can be divided into two main features (Ding et al.2016). Firstly, there are the near-vertical, densely packed fringes, which are visible both in figure3.2aand in figure3.2c. Secondly, there are the hyperbolic fringes in figure3.2c, slowly converging towards the nearby 3p resonance as τ increases.
Induced Attosecond Beatings The cause of the near-vertical fringes can be understood by considering the paths in the right side of figure3.3: At the time t = 0, the1Po spectrum – notably near the energies ε3p and εFof the 2s−13p resonance and a final state ψF
E– is populated using the XUV component of the pump. The system is then allowed to propagate freely until the arrival of the probe. During this time the phase difference between the states oscillates with a frequency of εF− ε3p . When the probe arrives, population is transferred from the resonance to ψFE
via two-photon stimulated emission, interfering either constructively or destructively depending on the relative phases of the population. This causes beatings of frequency εF− ε3p ≈ 2ωIR as τ changes.
These beatings are the cause of the local maximum at ωτ≈ 2ωIRin figure3.2b.
3.1. ATTOSECOND TRANSIENT ABSORPTION SPECTROSCOPY
3s 3s
3p 3p
1Se 1Po 1De 1.55
1.60 1.65 1.70
L ε [au]
0 τ
εF
ε3p ε3s
t [au]
ε [au]
Figure 3.3: To the left schematic illustration of the autoionising continuum, including the 2s−13s and 2s−13p resonances (labeled as 3s and 3p in the figure), below the second threshold of Neon, is shown. Approximate one- and two-photon beatings (illustrated with arrows to the right) induced between the 2s−13p state and lower states using an IR of frequency ωIR= 0.057117 au are shown with black arrows. The righthand figure show different paths excited electron population can follow over time during ATAS. The pump is centered at t = 0, and the probe at t = τ. Dashed paths are only significantly populated if the pump contains a low-energy component.
Due to selection rules, however, only beatings with an even number k of probe photons can be seen when the pump only populates the L = 1 spectrum.
The dashed arrows in the aforementioned figure show paths which are signif- icantly populated by adding a low-frequency component to the pump. This populates the 2s−13s state. At the time of the arrival of the probe, part of this population is transferred to ψF
Eand the 2s−13p resonance via one-photon absorption or stimulated emission. Analogously to the case mentioned above, this causes beatings of frequency |εF− ε3s| ≈ ωIR. These beatings are visible in figures3.2b(around ωτ= ωIR) and3.2d(slightly above ωτ= ωIRfor ωr= ε3p2), in the studied energy ranges.
2A peak is also present at ωτslightly below ωIR. This is due to the one-photon coupling with the
CHAPTER 3. AB-INITIO THEORY
To generalise the discussion above, the k−photon beating with a given reso- nance of energy εrescan be written as
ωbeat= |ωr− εres| ≈ kωIR, (3.38) at the response energy ωr.
Thus, the frequency of the beatings decreases with the energy-distance to the relevant resonance. This can be seen in figures3.2band3.2d: For all the beatings discussed above, an associated local maxima can be found, following a path where ωτdecreases linearly with |ωr− εres|. The same beatings are visible in ˜σ (ωr;ωτ) for negative values of ωτ, at ωτ= −|ωr− εres|.
The above reasoning assumes that the energy eigenstates are unperturbed by the external field. However, due to the dressing by the field, the state energies may experience a Stark (1913a,b) shift during the population transfers. The strength of such shifts can be measured by observing the deviation in ωτ of the localised ˜σ (ωr;ωτ)-maxima, from what would be predicted by the above discussion. (Argenti et al.2015; Freeman et al.1978,1987,1991; Ott et al.2013;
Petersson et al.2017)
The Buildup of Fano Profiles The second feature to be treated is that of the hyperbolic fringes converging towards the 2s−13p state in figure3.2c. Such fringes are not unique to that resonance, but rather a general trait of resonances populated during ATAS (Argenti et al.2015; Cheng et al. 2016; Ding et al.
2016).
The cause of such fringes is easily understood with the help of figure3.4. Fig- ure3.4ashows the (approximate, see figure text) population of the 2s−13pE
state for different values of τ (and for no probe), and figure3.4bthe corresponding spectrum near the 2s−13p−resonance to the given time-delay.
First, consider the case when no pulse is used. This is shown in black. At times t ≈ 0, the mixed XUV-IR pump populates the 2s−13pE
state. After the
